Integrand size = 10, antiderivative size = 80 \[ \int x^3 \cot ^{-1}(a x)^2 \, dx=\frac {x^2}{12 a^2}-\frac {x \cot ^{-1}(a x)}{2 a^3}+\frac {x^3 \cot ^{-1}(a x)}{6 a}-\frac {\cot ^{-1}(a x)^2}{4 a^4}+\frac {1}{4} x^4 \cot ^{-1}(a x)^2-\frac {\log \left (1+a^2 x^2\right )}{3 a^4} \]
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Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4947, 5037, 272, 45, 4931, 266, 5005} \[ \int x^3 \cot ^{-1}(a x)^2 \, dx=-\frac {\cot ^{-1}(a x)^2}{4 a^4}-\frac {x \cot ^{-1}(a x)}{2 a^3}+\frac {x^2}{12 a^2}-\frac {\log \left (a^2 x^2+1\right )}{3 a^4}+\frac {1}{4} x^4 \cot ^{-1}(a x)^2+\frac {x^3 \cot ^{-1}(a x)}{6 a} \]
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Rule 45
Rule 266
Rule 272
Rule 4931
Rule 4947
Rule 5005
Rule 5037
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \cot ^{-1}(a x)^2+\frac {1}{2} a \int \frac {x^4 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx \\ & = \frac {1}{4} x^4 \cot ^{-1}(a x)^2+\frac {\int x^2 \cot ^{-1}(a x) \, dx}{2 a}-\frac {\int \frac {x^2 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a} \\ & = \frac {x^3 \cot ^{-1}(a x)}{6 a}+\frac {1}{4} x^4 \cot ^{-1}(a x)^2+\frac {1}{6} \int \frac {x^3}{1+a^2 x^2} \, dx-\frac {\int \cot ^{-1}(a x) \, dx}{2 a^3}+\frac {\int \frac {\cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a^3} \\ & = -\frac {x \cot ^{-1}(a x)}{2 a^3}+\frac {x^3 \cot ^{-1}(a x)}{6 a}-\frac {\cot ^{-1}(a x)^2}{4 a^4}+\frac {1}{4} x^4 \cot ^{-1}(a x)^2+\frac {1}{12} \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )-\frac {\int \frac {x}{1+a^2 x^2} \, dx}{2 a^2} \\ & = -\frac {x \cot ^{-1}(a x)}{2 a^3}+\frac {x^3 \cot ^{-1}(a x)}{6 a}-\frac {\cot ^{-1}(a x)^2}{4 a^4}+\frac {1}{4} x^4 \cot ^{-1}(a x)^2-\frac {\log \left (1+a^2 x^2\right )}{4 a^4}+\frac {1}{12} \text {Subst}\left (\int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = \frac {x^2}{12 a^2}-\frac {x \cot ^{-1}(a x)}{2 a^3}+\frac {x^3 \cot ^{-1}(a x)}{6 a}-\frac {\cot ^{-1}(a x)^2}{4 a^4}+\frac {1}{4} x^4 \cot ^{-1}(a x)^2-\frac {\log \left (1+a^2 x^2\right )}{3 a^4} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.76 \[ \int x^3 \cot ^{-1}(a x)^2 \, dx=\frac {a^2 x^2+2 a x \left (-3+a^2 x^2\right ) \cot ^{-1}(a x)+3 \left (-1+a^4 x^4\right ) \cot ^{-1}(a x)^2-4 \log \left (1+a^2 x^2\right )}{12 a^4} \]
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Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.88
method | result | size |
parallelrisch | \(-\frac {-3 a^{4} x^{4} \operatorname {arccot}\left (a x \right )^{2}-2 a^{3} x^{3} \operatorname {arccot}\left (a x \right )-a^{2} x^{2}+6 \,\operatorname {arccot}\left (a x \right ) a x +1+3 \operatorname {arccot}\left (a x \right )^{2}+4 \ln \left (a^{2} x^{2}+1\right )}{12 a^{4}}\) | \(70\) |
parts | \(\frac {x^{4} \operatorname {arccot}\left (a x \right )^{2}}{4}+\frac {\frac {a^{3} x^{3} \operatorname {arccot}\left (a x \right )}{3}-\operatorname {arccot}\left (a x \right ) a x +\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )+\frac {a^{2} x^{2}}{6}-\frac {2 \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {\arctan \left (a x \right )^{2}}{2}}{2 a^{4}}\) | \(76\) |
derivativedivides | \(\frac {\frac {a^{4} x^{4} \operatorname {arccot}\left (a x \right )^{2}}{4}+\frac {a^{3} x^{3} \operatorname {arccot}\left (a x \right )}{6}-\frac {\operatorname {arccot}\left (a x \right ) a x}{2}+\frac {\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )}{2}+\frac {a^{2} x^{2}}{12}-\frac {\ln \left (a^{2} x^{2}+1\right )}{3}+\frac {\arctan \left (a x \right )^{2}}{4}}{a^{4}}\) | \(78\) |
default | \(\frac {\frac {a^{4} x^{4} \operatorname {arccot}\left (a x \right )^{2}}{4}+\frac {a^{3} x^{3} \operatorname {arccot}\left (a x \right )}{6}-\frac {\operatorname {arccot}\left (a x \right ) a x}{2}+\frac {\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )}{2}+\frac {a^{2} x^{2}}{12}-\frac {\ln \left (a^{2} x^{2}+1\right )}{3}+\frac {\arctan \left (a x \right )^{2}}{4}}{a^{4}}\) | \(78\) |
risch | \(-\frac {\left (a^{4} x^{4}-1\right ) \ln \left (i a x +1\right )^{2}}{16 a^{4}}+\frac {\left (3 i \pi \,a^{4} x^{4}+3 x^{4} \ln \left (-i a x +1\right ) a^{4}+2 i a^{3} x^{3}-6 i a x -3 \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{24 a^{4}}-\frac {i \pi \,x^{4} \ln \left (-i a x +1\right )}{8}+\frac {\pi ^{2} x^{4}}{16}-\frac {x^{4} \ln \left (-i a x +1\right )^{2}}{16}-\frac {i x^{3} \ln \left (-i a x +1\right )}{12 a}+\frac {\pi \,x^{3}}{12 a}+\frac {i x \ln \left (-i a x +1\right )}{4 a^{3}}+\frac {x^{2}}{12 a^{2}}-\frac {\pi x}{4 a^{3}}+\frac {\pi \arctan \left (a x \right )}{4 a^{4}}+\frac {\ln \left (-i a x +1\right )^{2}}{16 a^{4}}-\frac {\ln \left (a^{2} x^{2}+1\right )}{3 a^{4}}\) | \(224\) |
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Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75 \[ \int x^3 \cot ^{-1}(a x)^2 \, dx=\frac {a^{2} x^{2} + 3 \, {\left (a^{4} x^{4} - 1\right )} \operatorname {arccot}\left (a x\right )^{2} + 2 \, {\left (a^{3} x^{3} - 3 \, a x\right )} \operatorname {arccot}\left (a x\right ) - 4 \, \log \left (a^{2} x^{2} + 1\right )}{12 \, a^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98 \[ \int x^3 \cot ^{-1}(a x)^2 \, dx=\begin {cases} \frac {x^{4} \operatorname {acot}^{2}{\left (a x \right )}}{4} + \frac {x^{3} \operatorname {acot}{\left (a x \right )}}{6 a} + \frac {x^{2}}{12 a^{2}} - \frac {x \operatorname {acot}{\left (a x \right )}}{2 a^{3}} - \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{3 a^{4}} - \frac {\operatorname {acot}^{2}{\left (a x \right )}}{4 a^{4}} & \text {for}\: a \neq 0 \\\frac {\pi ^{2} x^{4}}{16} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96 \[ \int x^3 \cot ^{-1}(a x)^2 \, dx=\frac {1}{4} \, x^{4} \operatorname {arccot}\left (a x\right )^{2} + \frac {1}{6} \, a {\left (\frac {a^{2} x^{3} - 3 \, x}{a^{4}} + \frac {3 \, \arctan \left (a x\right )}{a^{5}}\right )} \operatorname {arccot}\left (a x\right ) + \frac {a^{2} x^{2} + 3 \, \arctan \left (a x\right )^{2} - 4 \, \log \left (a^{2} x^{2} + 1\right )}{12 \, a^{4}} \]
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\[ \int x^3 \cot ^{-1}(a x)^2 \, dx=\int { x^{3} \operatorname {arccot}\left (a x\right )^{2} \,d x } \]
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Time = 0.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.82 \[ \int x^3 \cot ^{-1}(a x)^2 \, dx=\frac {x^4\,{\mathrm {acot}\left (a\,x\right )}^2}{4}-\frac {\frac {\ln \left (a^2\,x^2+1\right )}{3}-\frac {a^2\,x^2}{12}+\frac {{\mathrm {acot}\left (a\,x\right )}^2}{4}-\frac {a^3\,x^3\,\mathrm {acot}\left (a\,x\right )}{6}+\frac {a\,x\,\mathrm {acot}\left (a\,x\right )}{2}}{a^4} \]
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